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Progress towards UFDs (#88)

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Patrick Stevens
2019-12-08 11:18:39 +00:00
committed by GitHub
parent 99c38495ce
commit 33098e94b0
20 changed files with 462 additions and 253 deletions
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29
Rings/PrincipalIdealDomains/Lemmas.agda
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@@ -20,15 +20,21 @@ open import Rings.Ideals.Lemmas R
open import Rings.Units.Definition R
open import Rings.Irreducibles.Lemmas intDom
open import Rings.Units.Lemmas R
open import Rings.Ideals.Prime.Definition {R = R}
open import Rings.Ideals.Prime.Lemmas {R = R}
open import Rings.Primes.Definition intDom
open import Rings.Primes.Lemmas intDom
open import Rings.Ideals.Principal.Lemmas R
open import Rings.Ideals.Maximal.Lemmas {R = R}
open Ring R
open Setoid S
open Equivalence eq
irreducibleImpliesMaximalIdeal : (r : A) → Irreducible r → {d : _} → MaximalIdeal (generatedIdeal r) {d}
MaximalIdeal.notContained (irreducibleImpliesMaximalIdeal r irred {d}) = 1R
MaximalIdeal.notContainedIsNotContained (irreducibleImpliesMaximalIdeal r irred {d}) = Irreducible.nonunit irred
MaximalIdeal.isMaximal (irreducibleImpliesMaximalIdeal r irred {d}) {biggerPred} bigger biggerContains (outsideR , (biggerContainsOutside ,, notInR)) {x} = biggerPrincipal' (unitImpliesGeneratedIdealEverything w {x})
irreducibleImpliesMaximalIdeal : {r : A} → Irreducible r → {d : _} → MaximalIdeal (generatedIdeal r) {d}
MaximalIdeal.notContained (irreducibleImpliesMaximalIdeal {r} irred {d}) = 1R
MaximalIdeal.notContainedIsNotContained (irreducibleImpliesMaximalIdeal {r} irred {d}) = Irreducible.nonunit irred
MaximalIdeal.isMaximal (irreducibleImpliesMaximalIdeal {r} irred {d}) {biggerPred} bigger biggerContains (outsideR , (biggerContainsOutside ,, notInR)) {x} = biggerPrincipal' (unitImpliesGeneratedIdealEverything w {x})
where
biggerGen : A
biggerGen = PrincipalIdeal.generator (pid bigger)
@@ -51,3 +57,18 @@ MaximalIdeal.isMaximal (irreducibleImpliesMaximalIdeal r irred {d}) {biggerPred}
v r|bg | assoc = notInR (associateImpliesGeneratedIdealsEqual' assoc (PrincipalIdeal.genGenerates (pid bigger) biggerContainsOutside))
w : Unit biggerGen
w = dividesIrreducibleImpliesUnit irred u v
primeIdealIsMaximal : {c : _} {pred : A → Set c} → {i : Ideal pred} → (nonzero : Sg A (λ a → ((a ∼ 0R) → False) && pred a)) → PrimeIdeal i → {d : _} → MaximalIdeal i {d}
primeIdealIsMaximal {pred = pred} {i} (m , (m!=0 ,, predM)) prime {d = d} = maximalIdealWellDefined (generatedIdeal (PrincipalIdeal.generator princ)) i (memberDividesImpliesMember i (PrincipalIdeal.genIsInIdeal princ)) (PrincipalIdeal.genGenerates princ) isMaximal
where
princ : PrincipalIdeal i
princ = pid i
isPrime : Prime (PrincipalIdeal.generator princ)
isPrime = primeIdealImpliesPrime (λ gen=0 → PrimeIdeal.notContainedIsNotContained prime (exFalso (m!=0 (generatorZeroImpliesAllZero princ gen=0 predM)))) (primeIdealWellDefined i (generatedIdeal (PrincipalIdeal.generator princ)) (PrincipalIdeal.genGenerates princ) (memberDividesImpliesMember i (PrincipalIdeal.genIsInIdeal princ)) prime)
isIrreducible : Irreducible (PrincipalIdeal.generator princ)
isIrreducible = primeIsIrreducible isPrime
isMaximal : MaximalIdeal (generatedIdeal (PrincipalIdeal.generator princ)) {d}
isMaximal = irreducibleImpliesMaximalIdeal isIrreducible {d}
irreducibleImpliesPrime : {x : A} → Irreducible x → Prime x
irreducibleImpliesPrime {x} irred = primeIdealImpliesPrime (Irreducible.nonzero irred) (idealMaximalImpliesIdealPrime (generatedIdeal x) (irreducibleImpliesMaximalIdeal irred))